1. Field of the Invention
The present invention relates to an electromagnetic wave analyzer and program which analyze electromagnetic waves by solving Maxwell""s equations in the time and spatial domains with finite difference methods. More particularly, the present invention relates to an electromagnetic wave analyzer and program which divide a computational domain into non-uniform cells to solve Maxwell""s equations.
2. Description of the Related Art
Today""s computational electromagnetics exploits the finite-difference time-domain (FD-TD) method as a technique to analyze the transitional behavior of electromagnetic waves, using computers for numerical calculation. The FD-TD algorithm solves Maxwell""s equations in time and spatial domains with finite difference methods. Because of its wide scope of applications and improved accuracy and efficiency, FD-TD solvers are used in various situations in these years. For details of FD-TD methods, see the following literatures: K. S. Yee, xe2x80x9cNumerical solution of initial boundary value problems involving Maxwell""s equations in isotropic media,xe2x80x9d IEEE Trans. Antennas and Propagation, Vol. 14, pp. 302-307, 1966; and A. Taflove, xe2x80x9cComputational Electrodynamics,xe2x80x9d MA, Artech House, 1995.
As mentioned above, the FD-TD algorithm solves Maxwell""s equations including two rotations, with difference methods in both the time domain (t) and spatial domain (x, y, z). The next part will present specific Maxwell""s equations for some different numbers of spatial dimensions. The following notation will be used to describe each problem:
Ex, Ey, Ez: electric field components in the x-, y-, and z-axis directions
Hx, Hy, Hz: magnetic field components in the x-, y-, and z-axis directions
xcexc: permeability
∈: dielectric constant
"sgr": electrical conductivity
First, in a one-dimensional space, the following two equations express how the electromagnetic waves propagate in the x-axis direction.                               μ          ⁢                                    ∂                              H                y                                                    ∂              t                                      =                              ∂                          E              z                                            ∂            x                                              (        1        )                                                      ϵ            ⁢                                          ∂                                  E                  z                                                            ∂                t                                              +                      σ            ⁢                          xe2x80x83                        ⁢                          E              z                                      =                              ∂                          H              y                                            ∂            x                                              (        2        )            
In a two-dimensional space, the transverse magnetic (TM) waves propagating in the x-axis and y-axis directions are expressed as follows.                               μ          ⁢                                    ∂                              H                x                                                    ∂              t                                      =                  -                                    ∂                              E                z                                                    ∂              y                                                          (        3        )                                          μ          ⁢                                    ∂                              H                y                                                    ∂              t                                      =                              ∂                          E              z                                            ∂            x                                              (        4        )                                                      ϵ            ⁢                                          ∂                                  E                  z                                                            ∂                t                                              +                      σ            ⁢                          xe2x80x83                        ⁢                          E              z                                      =                                            ∂                              H                y                                                    ∂              x                                -                                    ∂                              H                x                                                    ∂              y                                                          (        5        )            
Similarly, the following partial differential equations give the behavior of transverse electric (TE) waves in a two-dimensional space.                                           ϵ            ⁢                                          ∂                                  E                  x                                                            ∂                t                                              +                      σ            ⁢                          xe2x80x83                        ⁢                          E              x                                      =                              ∂                          H              z                                            ∂            y                                              (        6        )                                                      ϵ            ⁢                                          ∂                                  E                  y                                                            ∂                t                                              +                      σ            ⁢                          xe2x80x83                        ⁢                          E              y                                      =                  -                                    ∂                              H                z                                                    ∂              x                                                          (        7        )                                          μ          ⁢                                    ∂                              H                z                                                    ∂              t                                      =                                            ∂                              E                x                                                    ∂              y                                -                                    ∂                              E                y                                                    ∂              x                                                          (        8        )            
Lastly, electromagnetic waves in a three-dimensional space are expressed in the systems of six coupled partial differential equations as follows.                                           ϵ            ⁢                                          ∂                                  E                  x                                                            ∂                t                                              +                      σ            ⁢                          xe2x80x83                        ⁢                          E              x                                      =                                            ∂                              H                z                                                    ∂              y                                -                                    ∂                              H                y                                                    ∂              z                                                          (        9        )                                                      ϵ            ⁢                                          ∂                                  E                  y                                                            ∂                t                                              +                      σ            ⁢                          xe2x80x83                        ⁢                          E              y                                      =                                            ∂                              H                x                                                    ∂              z                                -                                    ∂                              H                z                                                    ∂              x                                                          (        10        )                                                      ϵ            ⁢                                          ∂                                  E                  z                                                            ∂                t                                              +                      σ            ⁢                          xe2x80x83                        ⁢                          E              z                                      =                                            ∂                              H                y                                                    ∂              x                                -                                    ∂                              H                x                                                    ∂              y                                                          (        11        )                                          μ          ⁢                                    ∂                              H                x                                                    ∂              t                                      =                                            ∂                              E                y                                                    ∂              z                                -                                    ∂                              E                z                                                    ∂              y                                                          (        12        )                                          μ          ⁢                                    ∂                              H                y                                                    ∂              t                                      =                                            ∂                              E                z                                                    ∂              x                                -                                    ∂                              E                x                                                    ∂              z                                                          (        13        )                                          μ          ⁢                                    ∂                              H                z                                                    ∂              t                                      =                                            ∂                              E                x                                                    ∂              y                                -                                    ∂                              E                y                                                    ∂              x                                                          (        14        )            
The FD-TD method is applied to a given computational domain, which is normally discretized into small meshes or boxes, called xe2x80x9ccellsxe2x80x9d or xe2x80x9cdifferencing grids.xe2x80x9d The size of those cells is one of the major factors that govern computational errors in FD-TD analysis. Electromagnetic waves may exhibit a sudden change in their transitional behavior in some particular region of the computational domain, often resulting in increased numerical errors. Such errors, however, can be reduced to an acceptable level if the selected cell size is sufficiently small.
Obviously, the simplest way of spatial discretization is to divide a given computational domain into evenly spaced grids. This uniform discretization, however, is likely to produce a large number of cells, thus consuming more computational resources including CPU time and memory space. To optimize the analytical model of interest, researchers often divide a computational domain into differently sized cells. This non-uniform discretization technique reduces the total number of cells, effectively alleviating the processing burden on the computer.
However, such a non-uniform computational domain has a known problem that the numerical error would increase at a boundary between two regions having different cell sizes. Such boundaries are referred to herein as the xe2x80x9ccell-size boundaries.xe2x80x9d Since the spatial step size changes at a cell-size boundary, the midpoint between two adjacent cells does not lie on that cell-to-cell boundary, unlike the cases of uniform discretization. The consequent displacement of assumed calculation points aggravates the accuracy of finite difference approximation, resulting in increased truncation errors.
The central differencing technique is known to be second-order accurate for equally sized cells. However, the accuracy of this technique is reduced to the first order at cell-size boundaries because the assumption of spatial uniformity fails in such particular space points. The resulting computational errors could sometimes reach a intolerable level for certain types of objects to be analyzed.
In view of the foregoing, it is an object of the present invention to provide an electromagnetic wave analyzer capable of handling non-uniform cells with smaller computational errors.
Another object of the present invention is to provide an electromagnetic wave analyzing program which divides a given computational domain into a plurality of cells and numerically analyzes the behavior of transitional electromagnetic fields on an individual cell basis, without introducing any additional errors.
To accomplish the first object, according to the present invention, there is provided an electromagnetic wave analyzer which divides a given computational domain into a plurality of cells and numerically analyzes transitional behavior of electromagnetic fields on an individual cell basis. This analyzer comprises the following elements: a cell size identification unit which identifies the uniformity of cells that surround each space point defined in the computational domain; a first calculation unit which calculates electromagnetic field components at a space point with a first calculation method when the cell size identification unit has identified the surrounding cells as being uniform in size; a second calculation unit which calculates electromagnetic field components at a space point with a second calculation method when the cell size identification unit has identified the surrounding cells as being non-uniform in size, where the second calculation method has smaller computational errors than those of the first calculation method; and a data output unit which outputs the electromagnetic field components at every space point which have been calculated by the first and second calculation units.
To accomplish the second object, the present invention provides a computer program product which enables a computer to divide a given computational domain into a plurality of cells and numerically analyze transitional behavior of electromagnetic fields on an individual cell basis. This program causes the computer to perform the steps of: (a) identifying uniformity of cells that surround each space point defined in the computational domain; (b) calculating electromagnetic field components at a space point with a first calculation method when the surrounding cells are identified as being uniform in size; (c) calculating electromagnetic field components at a space point with a second calculation method when the surrounding cells are identified as being non-uniform in size, the second calculation method having smaller computational errors than those of the first calculation method; and (d) outputting the electromagnetic field components at every space point which have been calculated in the first and second calculating steps (b) and (c).
The above and other objects, features and advantages of the present invention will become apparent from the following description when taken in conjunction with the accompanying drawings which illustrate preferred embodiments of the present invention by way of example.